Discrete Dynamics in Nature and Society

Volume 2016, Article ID 4863907, 14 pages

http://dx.doi.org/10.1155/2016/4863907

## Describing Urban Evolution with the Fractal Parameters Based on Area-Perimeter Allometry

^{1}Department of Geography, College of Urban and Environmental Sciences, Peking University, Beijing 100871, China^{2}Department of Urban Planning and Design, The University of Hong Kong, Pokfulam Road, Pokfulam, Hong Kong

Received 15 October 2015; Revised 10 December 2015; Accepted 10 December 2015

Academic Editor: Vicenç Méndez

Copyright © 2016 Yanguang Chen and Jiejing Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The area-perimeter allometric scaling is an important approach for researching fractal cities, and the basic ideas and models have been researched for a long time. However, the fractal parameters based on this scaling relation have not been efficiently utilized in urban studies. This paper is devoted to developing a description method of urban evolution using the fractal parameter sets based on the area-perimeter measure relation. The novelty of this methodology is as follows: first, the form dimension and boundary dimension are integrated to characterize the urban structure and texture; second, the global and local parameters are combined to characterize an urban system and individual cities; third, an entire analytical process based on the area-perimeter scaling is illustrated. Two discoveries are made in this work: first, a dynamic proportionality factor can be employed to estimate the local boundary dimension; second, the average values of the local fractal parameters are approximately equal to the corresponding global fractal parameters of cities. By illustrating how to carry out the area-perimeter scaling analysis of Chinese cities in Yangtze River Delta in the case of remote sensing images with low resolution, we propose a possible new approach to exploring fractal systems of cities.

#### 1. Introduction

A scientific study is always involved with two processes: the first process is to describe how a system works using mathematics or measurement, and the second process is to gain understanding of the causality behind the system using observation, experience, or artificially constructed experiments [1, 2]. Fractal geometry is a powerful tool of describing urban systems because fractal dimension is a kind of characteristic parameter to describe scale-free phenomena [3–9]. By fractal dimension description, we can get insight into the spatial dynamics of urban evolution [10–13]. There are two simple and convenient ways of understanding city fractals. One is measurement process (see, e.g., [14–19]), and the other is the geometric measure relation (see, e.g., [12, 20, 21]). If a measurement result such as boundary length, urban area, and land-use density depends on the scale to measure (the length of yardstick), we may face a fractal object. On the other hand, if the proportional relation between two measures such as urban area and urban perimeter is involved with a scaling exponent indicating a pair of fractional parameters, we will meet a fractal phenomenon. The former way can be used to realize any types of fractals, while the latter way can be employed to identify the fractal boundary of a region. Sometimes an urbanized area is treated as a Euclidean space, but the urban boundary can be regarded as a fractal line. In this case, the boundary dimension of a city provides a good way of looking at city size and shape [4]. So far, we have more than five empirical approaches to estimating the fractal dimension of geographical boundaries [22–27].

An urban area bears an analogy to a random Koch island; thus, the fractal dimension of its boundary can be estimated with the area-perimeter scaling [4, 22, 28, 29]. The geometrical measure relation between urban area and perimeter is in essence an allometric scaling relation, which is similar to the relation between urban area and population [30]. If the population size of a city is compared to the weight of an animal, then the urban area can be compared to the volume of the animal and the urban perimeter to the surface area (the area of the whole skin) [20]. In many cases, it is difficult to investigate city population, but it is easy to measure urban area and the corresponding perimeter by using the remote sensing images and the geographical information system (GIS) techniques. The area-perimeter allometric scaling is a simple approach to revealing the spatiotemporal evolution dynamics of urban systems. However, two problems remain to be solved. First, in many cases, it is impossible to compute the fractal dimension of each city in an urban system because of inadequate remote sensing data or the lower resolution of remote sensing images. Second, the traditional formula of the boundary dimension based on the area-perimeter scaling is not exact enough to guarantee the effective results.

The processes and patterns of urban evolution follow scaling laws [31–34]. Fractal geometry is an effective tool to explore scaling in cities. In recent years, the new formulae for revising the boundary dimension calculations through the area-perimeter scaling have been proposed [23]. By means of these formulae, we can correct the errors of the boundary dimension and estimate the form dimension of cities. Using the boundary dimension and form dimension, we can characterize urban shapes and the spatiotemporal evolution of urban systems. This paper is devoted to developing the description method of urban patterns and processes using the fractal dimension sets based on the area-perimeter allometric scaling. It tries to solve several problems as follows: how to combine the global fractal parameters with the local fractal parameters for spatial analysis; how to make use of remote sensing images of low resolution for urban fractal studies; how to understand the influence of urban sprawl on fractal dimension change. The rest of the work is organized as follows. In Section 2, several new formulae of fractal dimension estimation are introduced and clarified for spatiotemporal analysis of urban form and growth. In Section 3, the new models and formulae are applied to the system of cities and towns in Yangtze River Delta, China, to make an empirical analysis. In Section 4, based on the mathematical models and the empirical case, the new analytical process of urban evolution based on the urban area-perimeter scaling is presented and illustrated, and several related questions are discussed. Finally, the paper reaches its conclusions by outlining its major viewpoints.

#### 2. Fractal Parameters

##### 2.1. Basic Formulae

In theory, a city figure can be divided into two parts: one is the urban boundary, and the other is the urban area within the boundary. The former is termed* urban envelope* () and can be described with the boundary dimension [21], while the latter is named* urban area* () and can be characterized by the form dimension [4, 35]. The form dimension is a* structural dimension*, while the boundary dimension is a* textural dimension* [20, 36, 37]. The form dimension and boundary dimension compose the shape dimension of cities. In fact, the family of shape indexes includes the well-known form ratio, which is defined by area and perimeter of a city [38]. The form dimension can be measured by the area-scale relation, while the boundary dimension can be determined by the perimeter-scale relation [23]. In this sense, urban shape can be characterized by both the form and the boundary dimensions.

Based on digital maps or remote sensing images, fractal cities are always defined in a 2-dimensional space. In practice, the urban area can be regarded as a Euclidean plane with a dimension , and accordingly, the urban boundary is treated as a fractal line [4, 23]. Thus, the boundary dimension represents the basic fractal dimension of urban shape and can be estimated with the regression analysis based on the method of the ordinary least squares (OLS). By the fractal measure relation [28, 29, 39], the urban area and perimeter follow a power law as below:where refers to the perimeter of the urban envelope and refers to the corresponding urban area. As for the parameters, denotes the boundary dimension of traditional meaning and is related with the proportionality coefficient. In this context, the boundary dimension should be termed* initial boundary dimension* because it represents the conventional concept of the fractal dimension of urban boundary. Equation (1) is in fact an allometric scaling relation, and the scaling exponent iswhere denotes the Euclidean dimension of the embedding space of urban form. The boundary dimension is often estimated by the formula . From (1), it follows thatwhich is an approximate formula of the boundary dimension estimation. By analogy with squares and empirical analysis, the proportionality parameter is always taken as fixed value; that is, [40, 41]. In fact, the fixed coefficient can be replaced by a dynamic parameter, which can be estimated with regression analysis. Thus, (2) provides a simple approach to estimating the boundary dimension especially when spatial data are short for computing the fractal dimension.

##### 2.2. Adjusting Formulae

It is easy to evaluate the boundary dimension of a fractal region such as the Koch island and urban envelope [21, 36]. However, the method based on the geometric measure relation always overestimates the fractal dimension value [22, 30]. In order to lessen the errors resulting from (1) and (2), a formula is derived as follows [23]: where represents the* revised boundary dimension* of a city (a textural dimension). Equation (4) suggests a linear relation between and . If we calculate the value of using (3) or log-linear regression analysis, we can estimate the value by means of (4).

In reality, a city is a complex spatial system, and urban area does not correspond to a 2-dimensional region. In this case, (1) is not enough to describe the geometric measure relation between urban area and perimeter. According to the studies of Cheng [42], Imre [43], and Imre and Bogaert [44], (1) can be generalized as follows:where denotes the fractal dimension of urban form within the urban envelope (a structural dimension). The form dimension can be estimated with the following formula [23]:which suggests a hyperbolic relation between and . From (4) and (6), it follows thatwhich suggests a hyperbolic relation between and . Accordingly, . If , then we have , and* vice versa*. If so, we will have a Euclidean object. Based on (5), the allometric scaling exponent expressed by (2) can be rewritten as where refers to the revised scaling exponent of the area-perimeter allometry. An allometric exponent is usually a ratio of one fractal dimension to another fractal dimension. In many cases, it is allometric scaling exponents rather than fractal dimensions that play an important role in spatial analysis of urban systems [30, 45, 46].

Using the mathematical models, allometric scaling relations, and the fractal parameter formulae, we can describe and analyze the spatial development and evolution of urban systems in the real world. For an urban system, the fractal dimensions and the related allometric scaling exponents can be classified as global fractal parameters and local fractal parameters. The global fractal parameters can be reckoned with the cross-sectional data and used to describe a system of cities as a whole, while the local fractal parameters can be figured out by using the data of individual cities and used to describe each city as an element in the urban system.

#### 3. Empirical Analysis

##### 3.1. Study Area and Method

The area-perimeter scaling and the adjusting formulae of fractal dimension can be applied to the actual cities by means of remote sensing data. Yangtze River Delta in China is taken as a study area to make an empirical analysis. The region includes 68 cities and towns, which can be regarded as an urban system (Figure 1). The remote sensing images used in this research came from National Aeronautics and Space Administration (NASA), including Landsat MSS, TM, and ETM images from 1985, 1996, and 2005, which were downloaded from the Earth Science Data Interface (ESDI) at the Global Land Cover Facility (GLCF) center of University of Maryland (http://glcf.umd.edu/). Firstly, these images were transformed to the GCS_WGS_1984 geographic coordinate system and Asia_Lambert_Conformal_Conic projected coordinate system. Then, the supervised classification method (SCM) was employed to extract the built-up area and the boundary of each city in a given year. The results can be manually corrected in ESRI ArcGIS to ensure better accuracy. The SCM is one of the popular remote sensing image classification techniques [47]. The basic idea of this technique is to select “training areas” as representative samples to identify the land cover classes in an image. The classifier is then used to attach labels to all the image pixels according to the trained parameters. The maximum likelihood classification (MLC) is commonly used to generate the trained parameters, such as mean vectors and variance-covariance matrix. Then, the land cover is classified based on the spectral signature that is defined by these trained parameters. In this work, we use the software of ERDAS IMAGINE which has a well-defined classifier for the SCM. We classify the Landsat images into five classes of land cover: city, forest, agriculture, water, and other. The land cover of city can be obtained and then the rough boundary of cities can be extracted on the base of the classification.